1. Lecture II Non dissipative traps Evaporative cooling Bose-Einstein condensation

2. Phase space density From room temperature to 100  K n  = 10 -7 Magneto-optical trap Molasses 100  K 10  K Intrisically limited because of the dissipative character of the MOT.

3. No light, no heating due to absorption Relies on magnetic moment interaction The force results from the inhomogenity of the magnetic field Magnetic trapping (1) For an atom with an nuclear spin in the ground state

4. F=1,m=1 F=1,m=0 F=1,m=-1 z Maxwell's equations: No max of |B| in the vaccum. Atoms cannot be magnetically trapped in the lower energy state. Two-body inelastic collisions Three-body inelastic collisions (dimer Rb 2 ). Ultra High Vacuum chamber, backgound gas collisions. Photo: Bell Labs Local minimum of |B| + spin polarisation V=|  ||B| Non dissipative trap !!! Magnetic trapping (2)

5. Spin flips Majorana losses Magnetic trap: classical picture versus the quantum one Classically, the angle θ between the magnetic moment and the magnetic field is constant due to the rapid precession of µ around the magnetic field axis. Classical picture V=|  ||B| Quantum picture  can take only quantized values               F=2 F=1

6. Magnetic trap with coils What kind of gradient do we need ? Gradient scales as I/d 2 Magneto-optical trap: 1 mm, T=50  K  b' r = k B T b' = 10 Gauss/cm Atoms are further compressedb' ~ 200 Gauss/cm Two kind of solutions I ~ 1000 A, d ~ cm I ~ 0.1 A, d ~ 100 mm Microchip

7. Magnetic trap with coils One coil: B B 0 -2b'x b'y b'z x y z Two coils (antiHelmoltz): x y z O B -4b'x 2b'y 2b'z B 2 =4b' 2 (4x 2 +y 2 +z 2 )

8. Time averaged Orbital Potential (TOP) B -4b'x 2b'y 2b'z B 0 B 0 cos(  t) Quadupolar configuration O z x y Rotating field + =  trap  Larmor 100 Hz 5kHz 1 MHz

9. Ioffe pritchard trap depth: 1 mK constant bias field gradient curvature

10. Microchip traps Ioffe Pritchard traps of various aspect ratios: Y-shaped splitting and recombining regions. interferometry device

11. Atomic conveyer belt

12. Magnetic guide with wires

13. Magnetic guide with 4 tubes 2D Quadrupolar configuration x y B b'x -b'y Add a longitudinal bias field to avoid spin flips

14. Evaporation F=1,m=1 F=1,m=0 F=1,m=-1 z radio frequency wave Relies on the redistribution of energy through elastic collisions

15. Surface Evaporation J. Low Temp. Phys. 133, 229 (2003) works with silicon surface

16. Interactions between cold atoms Two-body problem: One-body scattering problem Scattering state (eigenstate of H with a positive energy) scattering amplitude At low energy, and if W decreases faster than r -3 at infinity: scattering length Two interaction potentials with the same scattering length lead to the same properties at sufficiently low temperature Exceptions: dipole-dipole interactions (magnetic or electric) 1/r interactions induced by laser (Kurizki et al)

17. Interactions between cold atoms Characteristic length from 0.1 to 10 nm W(r) r  varies rapidly with all parameters:  = number of bound states 75% 25% empirical law 75% 25% empirical law scattering length 0 0.020.04-0.04-0.02 a = 5 nm for 87 Rb

18. Evaporation: a simple model (1) 1) 2) 3) Infinite depth Finite depth Infinite depth harmonic confinement

19. Evaporation: a simple model (2) We deduce a power law dependence with The phase space density changes according to with and N: 10 9 10 6 T: 100  K 100 nK n  x 10 6 Typical numbers The real form of the potential only changes the exponent

20. Signature of condensation: time of flight 3 10 6 atoms in an anisotropic magnetic trap 100  m * 5  m 0,5 to 1  K Time of flight T > T c Boltzmann gas T < T c condensate isotropic expansion anisotropic expansion Camera CCD atoms Laser

21. 2001 Physcis Nobel Prize E. Cornell, W. Ketterle and C. Wieman "for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates" Bose Einstein condensation Review of Modern Physics, 74, 875 (2002); ibid 74, 1131 (2002)

22. Dipole trap gallery

23. Single atom in a dipole trap possible application in quantum computing ?

25. Is it possible to realize a continuous source of degenerate atoms ? PRL 93, 093003 (2004) 10 elastic collisions per atom First signal of evaporation and gain in phase space density